Question

If tangent and normal to the curve $y=2sinx+sin2x$ are drawn at $P(x=\frac{\pi }{3})$
then area of the quadrilateral formed by tangent, the normal at p and coordinate axes is

• A. $\frac{\pi }{3}$
• B. $3\pi$
• C. $\frac{\pi \sqrt{3}}{2}$
• D. $\pi$

Answer 1

Answer: (C)

$\frac{\pi \sqrt{3}}{2}$

$y^{\prime }\left(x\right)=2\cos x+2\cos 2x$

$y^{\prime }\left(\frac{\pi }{3}\right)=1-1=0$

$y\left(\frac{\pi }{3}\right)=\sqrt{3}+\frac{\sqrt{3}}{2}=3\frac{\sqrt{3}}{2}$

now, we are left with the axes and lines $y=3\frac{\sqrt{3}}{2}$ and $x=\frac{\pi }{3}$

the area of rectangle is,

$=xy$

$=\left(\frac{\pi }{3}\right)\left(3\frac{\sqrt{3}}{2}\right)$

$=\pi \frac{\sqrt{3}}{2}$